In this talk I will share results from my dissertation that demonstrate how undergraduate mathematics instructors who are committed to equitable and inclusive instruction can support students' success. I will share results from instructor and student interview data that describes specific instructional practices and the impacts these practices had on students' learning experiences. Areas of particular focus will include community building, use of flexible structures and resources, and student centered instructional strategies.
Supporting Student Success in Early Undergraduate Mathematics Courses
In June 2003 I gave a talk at the Annual Meeting of the Association for Symbolic Logic, University of Illinois at Chicago, on “An online database of classes of algebraic structures”. This list of mathematical structures is still on a website at http://math.chapman.edu/~jipsen/structures, but is mostly just an alphabetical list of links that point to (sometimes incomplete) axiomatic descriptions of about 300 categories of universal algebras. This past summer I started a project with Bianca Newell to recreate this list of (partially-ordered) algebraic structures as a computable LaTeX document that can be checked for consistency and updated more reliably than the previous collection of webpages. In this talk I will describe this project and recent joint work on partially ordered universal algebras with José Gil-Ferez. In this setting, a partially ordered universal algebra is a poset with finitary operations that are order-preserving or order-reversing in each argument, and congruences are replaced by compatible preorders. Our investigations are based on an unpublished paper from 2004 by Don Pigozzi: Partially ordered varieties and quasivarieties, available here [pdf].
We present combinatorial upper bounds on dimensions of certain imaginary root spaces for symmetric Kac-Moody algebras. These come from the realization of the corresponding infinity-crystal using quiver varieties. This results is an expression for the multiplicity as counting words subject to some conditions, somewhat similar to the known Lyndon word combinatorics. As with that framework counting these words is hard, but in our setup error is concentrated, and we obtain a tractable over-count giving a good estimate. The framework is general, but we only work out specifics in rank two and certain rank 3 cases. This includes joint work with Colin Williams and with Patrick Chan. (https://cuboulder.zoom.us/j/94126918788)
Power's 2-categorical pasting theorem, asserting that any pasting diagram in a 2-category has a unique composite, is at the basis of the 2-categorical graphical calculus, which is used extensively to develop the theory of 2-categories. In this talk we discuss an (\infty,2)-categorical analog of the pasting theorem, asserting that the space of composites of any pasting diagram in an (\infty,2)-category is contractible. This result, which is joint with Hackney—Ozornova—Riehl, rediscovers independent work by Columbus.
Supporting Student Success in Early Undergraduate Mathematics Courses